Subgroups — #26122491

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251807	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688893	C₂	1	2	yes	maximal
3	#20243127	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688892	C₂	1	2	yes	maximal
5	C₄₇×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689079	—	2	4	no	—
9	C₄×#10625329	—	2	2	no	—
10	#15688893	C₄	1	8	yes	complemented
11	#15688892	—	2	4	no	—
12	#15688892	—	2	1	no	—
13	#15688892	—	2	1	no	—
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#389	—	2	2	no	—
18	C₄₇×#385	C₄	1	4	yes	—
19	C₄₇×#384	—	2	1	no	—
20	C₄₇×#382	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	C₂×C₄	1	4	yes	—
30	C₂×#10625329	C₂×C₄	1	4	yes	—
31	C₂×#10625329	D₄	1	4	yes	—
32	C₂×#10625329	Q₈	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	#13093935	—	2	4	no	—
36	C₄×C₄₇⋊C₈	—	2	4	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	2	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	1	no	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	8	2	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	4	1	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₄⋊C₄×C₄₇	—	4	1	no	—
77	C₄⋊C₄×C₄₇	—	8	4	no	—
78	C₂²⋊C₄×C₄₇	—	2	2	no	—
79	C₂²⋊C₄×C₄₇	—	2	4	no	—
80	C₂²⋊C₄×C₄₇	C₄²	1	4	yes	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	4	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	4	2	no	—
90	#10625329	—	4	2	no	—
91	#10625329	C₄⋊C₄	1	8	yes	complemented
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	4	1	no	—
99	C₂³×C₄₇	—	4	1	no	—
100	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	#117	1	1	yes	characteristic
103	C₂³×C₄₇	C₄×D₄	1	2	yes	—
104	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	8	4	no	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#121	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#123	1	2	yes	—
127	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#125	1	2	yes	—
129	C₂×C₄×C₄₇	#126	1	2	yes	—
130	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
131	C₄₇⋊C₈	—	2	8	no	—
132	C₄₇⋊C₈	—	4	4	no	—
133	#10916	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	4	1	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₄₇	#386	1	1	yes	characteristic
144	C₂²×C₄₇	#387	1	2	yes	—
145	C₂²×C₄₇	#391	1	1	yes	characteristic
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	—	4	2	no	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#385	1	2	yes	—
151	C₄×C₄₇	#387	1	2	yes	—
152	C₄×C₄₇	#388	1	2	yes	—
153	C₄×C₄₇	#389	1	2	yes	—
154	#2873	—	47	1	no	—
155	#2317	D₄₇	1	1	yes	characteristic
156	C₂×#344	—	47	2	no	—
157	#1742	—	47	1	no	—
158	C₂×#343	—	47	2	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	#1436	1	1	yes	characteristic
164	C₂×C₄₇	#1914	1	1	yes	characteristic
165	C₂×C₄₇	#1915	1	1	yes	characteristic
166	C₂²×M₁₆	—	47	2	no	—
167	C₂²×M₁₆	—	47	2	no	—
168	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
171	#405	—	94	4	no	—
172	C₄×M₁₆	—	94	2	no	—
173	#389	—	2	2	no	—
174	#385	Dic₄₇	1	4	yes	—
175	#384	—	2	1	no	—
176	#382	—	2	1	no	—
177	#344	—	47	8	no	—
178	#343	—	94	4	no	—
179	#343	—	94	1	no	—
180	#343	—	94	1	no	—
181	C₄₇	#10916	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	47	4	no	—
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	47	4	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₄×D₄₇	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
214	#97	—	94	4	no	—
215	C₄×C₈	—	94	4	no	—
216	#95	—	2	4	no	—
217	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	8	2	no	—
222	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
225	C₂²×C₄	#10625350	1	2	yes	—
226	C₂²×C₄	D₄×D₄₇	1	2	yes	—
227	C₂²×C₄	#10625352	1	2	yes	—
228	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
229	C₂²×C₄	#10625354	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	M₁₆	—	47	8	no	—
233	M₁₆	—	94	4	no	—
234	M₁₆	—	94	4	no	—
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	4	no	—
237	M₁₆	—	94	4	no	—
238	M₁₆	—	188	2	no	—
239	M₁₆	—	188	2	no	—
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	188	2	no	—
245	C₂×C₈	—	188	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	8	4	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	C₄×Dic₄₇	1	4	yes	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
263	C₂³	#13093983	1	1	yes	characteristic
264	C₂³	D₄×Dic₄₇	1	2	yes	—
265	C₂³	Q₈×Dic₄₇	1	2	yes	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
277	C₂×C₄	#13093982	1	2	yes	—
278	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
279	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
280	C₂×C₄	#13093986	1	2	yes	—
281	C₂×C₄	#13093989	1	2	yes	—
282	C₂×C₄	#13093992	1	1	yes	characteristic
283	C₂×C₄	#13093996	1	2	yes	—
284	C₂×C₄	#13094000	1	2	yes	—
285	C₂×C₄	#13094002	1	2	yes	—
286	C₂×C₄	#13094005	1	1	yes	characteristic
287	C₂×C₄	#13094009	1	2	yes	—
288	C₂×C₄	#13094013	1	1	yes	characteristic
289	C₂×C₄	#13094014	1	2	yes	—
290	C₂×C₄	#13094019	1	2	yes	—
291	C₂×C₄	#13094022	1	1	yes	characteristic
292	C₈	—	94	8	no	—
293	C₈	—	188	4	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
302	C₂²	C₄⋊C₄×Dic₄₇	1	2	yes	—
303	C₂²	#15689003	1	1	yes	characteristic
304	C₂²	#15689009	1	2	yes	—
305	C₂²	#15689030	1	1	yes	characteristic
306	C₄	—	4	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	#15688994	1	2	yes	—
311	C₄	#15689004	1	2	yes	—
312	C₄	#15689013	1	2	yes	—
313	C₄	#15689018	1	2	yes	—
314	C₂	—	2	1	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#20241620	1	1	yes	characteristic
319	C₂	#20243936	1	1	yes	characteristic
320	C₂	#20243941	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689003
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2317	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2317	D₄₇
hypocenter	C₄₇	#10916
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. #15688893	C₄	semidirect
91. #10625329	C₄⋊C₄	semidirect
181. C₄₇	#10916	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20251807	2
C₂×#15688893	2
#20243127	2
C₂×#15688892	2
C₄₇×#2317	2
#10916	47